Ones that reproduce through their roots with their roots, and ones that produce seeds under ground?

Also, do both of these need to be pollinated with neighboring plants to do this, or are they strictly asexual?

For the ones that reproduce by underground (or at the ground surface) parts these parts would be called stolons, rhizomes, tubers, or bulbs, and this is asexual reproduction.
http://en.wikipedia.org/wiki/Stolon
http://en.wikipedia.org/wiki/Rhizome
http://www.backyardnature.net/stemtype.htm
When these reproduce by having the these parts divide or break off, there's no pllenation involved.

There are different types of plants that produce seeds underground, from some orchids that never appear above ground (hypogeous, for hypo- "under" and geo - "earth" or "soil") to ones like peanuts that are pollinated above ground, but the seed develops below ground. I don't know of any special horticultural name for that. The peanut is sexual reproduction. I'm not sure if the orchid is self-fertile (pollinated within the flower) or if insects or worms are involved in cross pollinating neighboring plants.